For linear partial differential equations with known fundamental solutions, this work introduces a novel operator learning framework that relies exclusively on domain boundary data, including solution values and normal derivatives, rather than full-domain sampling. By integrating the previously developed Mathematical Artificial Data (MAD) method, which enforces physical consistency, all training data are synthesized directly from the fundamental solutions of the target problems, resulting in a fully data-driven pipeline without the need for external measurements or numerical simulations. We refer to this approach as the Mathematical Artificial Data Boundary Neural Operator (MAD-BNO), which learns boundary-to-boundary mappings using MAD-generated Dirichlet-Neumann data pairs. Once trained, the interior solution at arbitrary locations can be efficiently recovered through boundary integral formulations, supporting Dirichlet, Neumann, and mixed boundary conditions as well as general source terms. The proposed method is validated on benchmark operator learning tasks for two-dimensional Laplace, Poisson, and Helmholtz equations, where it achieves accuracy comparable to or better than existing neural operator approaches while significantly reducing training time. The framework is naturally extensible to three-dimensional problems and complex geometries.

翻译：针对已知基本解的线性偏微分方程，本文提出了一种新颖的算子学习框架，该框架完全依赖于域边界数据（包括解值和法向导数），而非全域采样。通过整合先前开发的、用于保证物理一致性的数学人工数据方法，所有训练数据均直接从目标问题的基本解合成，从而形成一个完全数据驱动的流程，无需外部测量或数值模拟。我们将该方法称为数学人工数据边界神经算子，它利用MAD生成的狄利克雷-诺伊曼数据对学习边界到边界的映射关系。一旦训练完成，即可通过边界积分公式高效地恢复任意位置的内域解，该方法支持狄利克雷、诺伊曼及混合边界条件，并能处理一般源项。所提方法在二维拉普拉斯方程、泊松方程和亥姆霍兹方程的基准算子学习任务上得到验证，其精度达到或优于现有神经算子方法，同时显著减少了训练时间。该框架可自然地扩展至三维问题及复杂几何情形。